A planet’s climate can be strongly affected by its orbital eccentricity and obliquity. We used a one-dimensional energy balance model modified to include a simple runaway greenhouse (RGH) parameterization to explore the effects of these two parameters on the climate of Earth-like aqua planets orbiting F-, G-, K-, and M-dwarf stars. We find that the range of instellations for which planets exhibit habitable surface conditions throughout an entire orbit decreases with increasing eccentricity. The appearance of temporarily habitable conditions during an orbit creates an eccentric habitable zone (EHZ) that is sensitive to orbital eccentricity, obliquity, and host star spectral type. The fraction of an orbit over which habitable surface conditions exist is larger on M-dwarf planets, due to their lower broadband planetary albedos. Larger planetary obliquities produce greater areas of surface habitability, yet generate smaller overall EHZs. We calculate the water mass loss rates of our simulated planets, and find that no transient RGH state exists on planets at all eccentricities. Rather, planets spend their entire orbits either in an RGH or not. G-dwarf planets receiving 100% of the modern solar constant will enter a RGH state at eccentricities larger than 0.55, leading to complete desiccation of the planetary surface in 3.6 Gyr, assuming an Earth-like water inventory. Similarly, M-dwarf planets can become desiccated in as little as 690 Myr for eccentricities above 0.38. This work has important implications for eccentric planets that may exhibit surface habitability despite technically departing from the traditional habitable zones of their host stars.